I currently have impulse response diagrams for first-order Taylor expansion and second-order Taylor expansion.
But the variable L that I am most concerned about has a shape similar to П under first-order conditions, which does not meet my expectations.
When I use second-order expansion, its shape first fluctuates positively and negatively, and then converges to the steady-state level. The latter seems to be more in line with my expectations. And the equation for L is an exponential nonlinear equation.
So what I want to ask is, in such a situation where there is a huge difference between first-order and second-order, should I choose second-order? Will the contradiction between the first and second order become the point of contradiction in my article?
Second-order effects are typically, by their nature, of second-order importance. If your model shows unintuitive behavior at first order, that is something you should investigate.
I compared the difference between first-order expansion and second-order expansion, and only the graph of the variable labor is completely different. In the system of equations, only the labor supply equation (which is also the variable I am most concerned about) appears in exponential form, while the other equations are basically linear equations. I wonder if this is the reason? If I determine that this is the reason, what is my next goal? And does this need to be explained in the article?
Yes, that would explain why mostly labor is affected when you move to higher order. But it does not explain why you get such large second-order effects in the first place. Do you consider (implausibly) large shocks? And why is only that one equation nonlinear while the others are linear?